On the Skew Laplacian Energy of a Digraph 1 INTRODUCTION
On the Skew Laplacian Energy of a Digraph 1 INTRODUCTION
On the Skew Laplacian Energy of a Digraph 1 INTRODUCTION
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International Ma<strong>the</strong>matical Forum, 4, 2009, no. 39, 1907 - 1914<br />
<strong>On</strong> <strong>the</strong> <strong>Skew</strong> <strong>Laplacian</strong> <strong>Energy</strong><br />
<strong>of</strong> a <strong>Digraph</strong><br />
C. Adiga and M. Smitha<br />
Department <strong>of</strong> Studies in Ma<strong>the</strong>matics<br />
University <strong>of</strong> Mysore, Manasagangotri<br />
Mysore-570006, India<br />
c adiga@hotmail.com<br />
smithamahadev@yahoo.co.in<br />
Abstract<br />
In this paper we introduce <strong>the</strong> concept <strong>of</strong> <strong>the</strong> skew <strong>Laplacian</strong> energy<br />
<strong>of</strong> a simple, connected digraph. We derive an explicit formula for <strong>the</strong><br />
skew <strong>Laplacian</strong> energy <strong>of</strong> a digraph G. We also find <strong>the</strong> minimal value<br />
<strong>of</strong> this energy in <strong>the</strong> class <strong>of</strong> all connected digraphs on n ≥ 2 vertices.<br />
Ma<strong>the</strong>matics Subject Classification: 05C50<br />
Keywords: <strong>Digraph</strong>s, skew energy, skew <strong>Laplacian</strong> energy<br />
1 <strong>INTRODUCTION</strong><br />
Let G be a simple (n, m) digraph with vertex set V (G) ={v1,v2, ..., vn}<br />
and arc set Γ(G) ⊂ V (G) × V (G). The skew-adjacency matrix <strong>of</strong> G is <strong>the</strong><br />
n × n matrix S(G) =[aij] where aij = 1 whenever (vi,vj) ∈ Γ(G), aij = −1<br />
whenever (vj,vi) ∈ Γ(G), aij = 0 o<strong>the</strong>rwise. Hence S(G) is a skew symmetric<br />
matrix <strong>of</strong> order n and all its eigen values are <strong>of</strong> <strong>the</strong> form iλ where i = √ −1<br />
and λ ∈ R. The skew energy <strong>of</strong> G is sum <strong>of</strong> <strong>the</strong> absolute values <strong>of</strong> eigenvalues<br />
<strong>of</strong> S(G). For additional informations on skew energy <strong>of</strong> digraphs we refer<br />
to [1]. Let D(G) =diag(d1,d2,d3, ..., dn) <strong>the</strong> digonal matrix with <strong>the</strong> vertex<br />
degrees d1,d2,d3, ..., dn <strong>of</strong> v1,v2, ..., vn. Then L(G) =D(G)−S(G) is called <strong>the</strong><br />
<strong>Laplacian</strong> matrix <strong>of</strong> <strong>the</strong> digraph G. Let λ1,λ2,λ3, ..., λn be <strong>the</strong> eigenvalues <strong>of</strong><br />
L(G). Then <strong>the</strong> set σSL(G) ={λ1,λ2,λ3, ..., λn} is called <strong>the</strong> skew <strong>Laplacian</strong><br />
spectrum <strong>of</strong> <strong>the</strong> digraph G. The <strong>Laplacian</strong> matrix <strong>of</strong> a simple, undirected graph<br />
G is D(G) − A(G) where A(G) is <strong>the</strong> adjacency matrix <strong>of</strong> G. It is symmetric,
1908 C. Adiga and M. Smitha<br />
singular, positive semi-definite and all its eigenvalues are real and non negative.<br />
It is well known that <strong>the</strong> smallest eigen value is zero and its multiplicity is equal<br />
to <strong>the</strong> number <strong>of</strong> connected components <strong>of</strong> G. The <strong>Laplacian</strong> spectrum <strong>of</strong> <strong>the</strong><br />
undirected graph G is <strong>the</strong> sum <strong>of</strong> squares <strong>of</strong> eigenvalues <strong>of</strong> its <strong>Laplacian</strong> matrix.<br />
For <strong>the</strong> results and background on <strong>the</strong> <strong>Laplacian</strong> spectrum we refer to [2],[3]<br />
and <strong>the</strong> references contained <strong>the</strong>rein.<br />
In this paper we will consider <strong>the</strong> problem <strong>of</strong> deriving a formula for ESL(G),<br />
<strong>the</strong> skew <strong>Laplacian</strong> energy <strong>of</strong> a digraph G interms <strong>of</strong> degrees <strong>of</strong> its vertices. A<br />
similar problem for <strong>the</strong> usual <strong>Laplacian</strong> energy has been considered in [4]. We<br />
also find <strong>the</strong> minimal value <strong>of</strong> ESL(G), in <strong>the</strong> class <strong>of</strong> all connected digraphs<br />
on n ≥ 2 vertices.<br />
2 FORMULA AND BOUNDS FOR THE SKEW<br />
LAPLACIAN ENERGY OF A<br />
DIGRAPH<br />
We begin by giving <strong>the</strong> formal definition <strong>of</strong> skew <strong>Laplacian</strong> energy.<br />
Definition 2.1 Let S(G) be <strong>the</strong> skew adjacency matrix <strong>of</strong> a simple digraph<br />
G. Then <strong>the</strong> skew <strong>Laplacian</strong> energy <strong>of</strong> <strong>the</strong> digraph G is defined as<br />
ESL(G) =<br />
where n is <strong>the</strong> order <strong>of</strong> G and λ1,λ2,λ3, ..., λn are <strong>the</strong> eigenvalues <strong>of</strong> <strong>the</strong> <strong>Laplacian</strong><br />
matrix L(G) =D(G) − S(G) <strong>of</strong> <strong>the</strong> digraph G.<br />
Example 2.2 Let G be a directed path on four vertices with <strong>the</strong> arc set<br />
(1,2)(2,3)(4,3).<br />
n<br />
i=1<br />
λ 2 i
<strong>On</strong> <strong>the</strong> skew <strong>Laplacian</strong> energy <strong>of</strong> a digraph 1909<br />
Then<br />
and so<br />
3<br />
2<br />
✲ ✲ ✛ <br />
1 2 3 4<br />
Fig.1 G = P4<br />
⎛<br />
0 1 0<br />
⎞<br />
0<br />
⎛<br />
1 0 0<br />
⎞<br />
0<br />
⎜<br />
S(G) = ⎜−1<br />
⎝ 0<br />
0<br />
−1<br />
1<br />
0<br />
0 ⎟<br />
−1⎠<br />
,<br />
⎜<br />
D(G) = ⎜0<br />
⎝0<br />
2<br />
0<br />
0<br />
2<br />
0 ⎟<br />
0⎠<br />
0 0 1 0<br />
0 0 0 1<br />
⎛<br />
1 −1 0<br />
⎞<br />
0<br />
⎜<br />
L(G) = ⎜1<br />
⎝0<br />
2<br />
1<br />
−1<br />
2<br />
0 ⎟<br />
1⎠<br />
0 0 −1 1<br />
.<br />
The eigenvalues <strong>of</strong> L(G) are 3 √ √ 1 1<br />
3 1 1<br />
√<br />
+ i + −4+2i, − i + −4 − 2i,<br />
2 2 2<br />
2 2 2<br />
−4 − 2i, and <strong>the</strong> skew <strong>Laplacian</strong> energy <strong>of</strong><br />
+ 1<br />
2<br />
i − 1<br />
2<br />
√ −4+2i, 3<br />
2<br />
<strong>the</strong> G is ESL(G) =4.<br />
− 1<br />
2<br />
i − 1<br />
2<br />
Example 2.3 Let G be a directed cycle on four vertices with <strong>the</strong> arc set<br />
{(1, 2), (2, 3), (3, 4), (4, 1)}.
1910 C. Adiga and M. Smitha<br />
Then<br />
L(G) =<br />
1 ✲ 2<br />
✻<br />
❄<br />
✛ <br />
4 3<br />
Fig.2. G = C4<br />
⎛<br />
⎜<br />
⎝<br />
2 −1 0 1<br />
1 2 −1 0<br />
0 1 2 −1<br />
−1 0 1 2<br />
The eigenvalues <strong>of</strong> L(G) are 2 + i √ 2, 2+i √ 2, 2 − i √ 2, 2 − i √ 2, and <strong>the</strong><br />
skew <strong>Laplacian</strong> energy <strong>of</strong> <strong>the</strong> G is ESL(G) =8.<br />
Let G1 =(V (G1), Γ(G1)) and G2 =(V (G2), Γ(G2)) be two finite, simple directed<br />
graphs with disjoint sets <strong>of</strong> vertices V (G1) and V (G2). Then <strong>the</strong> direct<br />
sum G = G1 ∔ G2 <strong>of</strong> <strong>the</strong>se graphs is defined by V (G) =V (G1) ∪ V (G2) and<br />
Γ(G) =Γ(G1) ∪ Γ(G2). Then we have <strong>the</strong> following <strong>the</strong>orem.<br />
Theorem 2.4. If G = G1 ∔ G2 is <strong>the</strong> direct sum <strong>of</strong> two finite, simple digraphs<br />
G1,G2, <strong>the</strong>n<br />
σSL(G) =σSL(G1) ∪ σSL(G2).<br />
⎞<br />
⎟<br />
⎠ .<br />
By Theorem 2.4 we immediately get <strong>the</strong> following <strong>the</strong>orem.<br />
Theorem 2.5. If G is a disconnected digraph whose components are G1,G2, ..., Gm,
<strong>On</strong> <strong>the</strong> skew <strong>Laplacian</strong> energy <strong>of</strong> a digraph 1911<br />
Then<br />
ESL(G) =<br />
m<br />
ESL(Gi).<br />
i=1<br />
Theorem 2.6. Let G be a simple digraph with vertex degrees are d1,d2, ..., dn.<br />
Then we have<br />
ESL(G) =<br />
n<br />
di(di − 1). (2.1)<br />
i=1<br />
Pro<strong>of</strong>. Let G be a simple digraph with vertex set V (G) ={v1,v2, ..., vn} and<br />
d(vi) =di for i =1, 2, ..., n. Let λ1,λ2,λ3, ..., λn be <strong>the</strong> eigenvalues <strong>of</strong> <strong>the</strong><br />
<strong>Laplacian</strong> matrix L(G) =D(G) − S(G) where D(G) =diag(d1,d2, ..., dn) and<br />
n<br />
S(G) is <strong>the</strong> skew-adjacency matrix <strong>of</strong> digraph G. We have λi = sum <strong>of</strong><br />
determinants <strong>of</strong> all 1 × 1 principal submatrices <strong>of</strong> L(G) = trace <strong>of</strong> L(G)<br />
n<br />
= di. Note that <br />
λiλj = sum <strong>of</strong> determinants <strong>of</strong> all 2 × 2 principal<br />
i=1<br />
submatrices <strong>of</strong> L(G)<br />
= <br />
det<br />
i
1912 C. Adiga and M. Smitha<br />
This completes <strong>the</strong> pro<strong>of</strong>.<br />
=<br />
=<br />
n<br />
i=1<br />
d 2 i −<br />
n<br />
i=1<br />
di<br />
n<br />
di(di − 1).<br />
i=1<br />
We remark that Theorem 2.6 shows that <strong>the</strong> skew <strong>Laplacian</strong> energy <strong>of</strong> a digraph<br />
is independnt <strong>of</strong> its orientation.<br />
Corollary 2.7. For any simple, digraph G, its skew <strong>Laplacian</strong> energy ESL(G)<br />
is an even integer.<br />
Pro<strong>of</strong>. Since di ≥ 1 for i =1, 2, ..., n, we have di(di − 1) is even for i =<br />
n<br />
1, 2, ..., n. Hence <strong>the</strong>ir sum di(di − 1) is also an even integer.<br />
i=1<br />
Corollary 2.8 If H is a proper subgraph <strong>of</strong> a connected digraph G with atleast<br />
three vertices, <strong>the</strong>n ESL(H) 0. Since H is obtained by removing atleast one edge from <strong>the</strong><br />
connected digraph G, using (2.1) we conclude that ESL(H)
<strong>On</strong> <strong>the</strong> skew <strong>Laplacian</strong> energy <strong>of</strong> a digraph 1913<br />
Pro<strong>of</strong>. Let G be any simple, connected digraph with n(≥ 2) vertices. Since<br />
<strong>the</strong> degree <strong>of</strong> any vertex is less than or equal to (n − 1), we have<br />
ESL(G) =<br />
n<br />
di(di − 1) ≤<br />
i=1<br />
n<br />
(n − 1)(n − 2) = n(n − 1)(n − 2).<br />
i=1<br />
It is also clear that <strong>the</strong> maximal skew <strong>Laplacian</strong> energy in <strong>the</strong> class <strong>of</strong> digraphs<br />
with n vertices is achieved for <strong>the</strong> complete digraph Kn. Also only <strong>the</strong> digraph<br />
Kn has this maximal energy.<br />
We prove <strong>the</strong> first inequality by induction. The result is obviously true<br />
for n = 2. Suppose <strong>the</strong> result is true for any connected digraph with n − 1<br />
vertices. We prove <strong>the</strong> result for any arbitrary connected digraph with n<br />
vertices. Let G be any connected digraph with n vertices. Then, <strong>the</strong>re is an<br />
induced subgraph H ⊂ G on (n − 1) vertices which is also connected. Denote<br />
V (H) ={v1,v2, ..., vn−1} and V (G) =V (H) ∪{vn}. Since G is connected, vn<br />
is adjacent to atleast one vertex vi(1 ≤ i ≤ n − 1). Assume that vn is adjacent<br />
to vn−1. Denote by G1 <strong>the</strong> graph with <strong>the</strong> same vertex set as G, induced<br />
in G by H and <strong>the</strong> pendent arc vn−1vn. Then we have H ⊂ G1 ⊂ G and<br />
ESL(G1) ≤ ESL(G). By <strong>the</strong> induction hypothisis we have 2n − 6 ≤ ESL(H).<br />
Moreover ESL(G) =ESL(H)+2d where d is <strong>the</strong> degree <strong>of</strong> vn−1 in <strong>the</strong> graph<br />
H. Thus<br />
ESL(G) ≥ ESL(G1) =ESL(H)+2d ≥ (2n − 6) + 2 = 2n − 4.<br />
Hence <strong>the</strong> first inequality in (2.2) is also true for G. By Corollary 2.9 we have<br />
ESL(Pn) =2n − 4. We will now prove that if G is a simple, connected digraph<br />
with n ≥ 2 vertices such that ESL(G) =2n − 4, <strong>the</strong>n G must be a directed<br />
path Pn. Again we prove this by induction on n. Let H and G1 have <strong>the</strong> same<br />
meaning as in <strong>the</strong> previous part <strong>of</strong> <strong>the</strong> pro<strong>of</strong>. We prove that dn = d(vn) =1.<br />
<strong>On</strong> <strong>the</strong> contrary, let us assume that dn ≥ 2 and let vn be adjacent not only to<br />
vn−1 but also to some vn−2 ∈ V (H). We have<br />
and<br />
Thus<br />
ESL(H) ≥ 2n − 6<br />
ESL(G) ≥ ESL(H)+2dH(vn−2)+2dH(vn−1)+2.<br />
2n − 4=ESL(G) ≥ (2n − 6)+2+2+2=2n,<br />
which is a contradiction. Hence dn = d(vn) = 1, so G is <strong>the</strong> same as G1. Since<br />
ESL(G) =ESL(G1) =ESL(H)+2dH(vn−1) =2n − 4,
1914 C. Adiga and M. Smitha<br />
we obtain<br />
This implies<br />
Therefore<br />
2dH(vn−1) =(2n − 4) − ESLH ≤ (2n − 4) − (2n − 6) = 2.<br />
dH(vn−1) =1.<br />
ESL(H) =(2n − 4) − 2=2n − 6.<br />
By <strong>the</strong> induction hypo<strong>the</strong>sis this means that H is <strong>the</strong> path Pn−1 and vn−1<br />
is end vertex <strong>of</strong> this path. This yields that G must be <strong>the</strong> path Pn. This<br />
completes <strong>the</strong> pro<strong>of</strong>.<br />
Acknowledgement: The first author is thankful to Department <strong>of</strong> Science<br />
and Technology, Government <strong>of</strong> India, New Delhi for <strong>the</strong> financial support<br />
under <strong>the</strong> grant DST/SR/S4/MS:490/07.<br />
References<br />
[1] C. Adiga, R. Balakrishnan and Wasin So, The skew energy <strong>of</strong> a graph<br />
(communicated for publication).<br />
[2] R. Grone, R. Merris, V.Sunder, The <strong>Laplacian</strong> spectrum <strong>of</strong> a graph, SIAM<br />
J.Matrix Anal. Appl. 11(1990), 218-238.<br />
[3] R. Merris, <strong>Laplacian</strong> matrices <strong>of</strong> graphs, A survey, Linear Algebra and its<br />
Appl. 197,198 (1994), 143-176.<br />
[4] Mirjana Lazic Kragujevac, <strong>On</strong> <strong>the</strong> <strong>Laplacian</strong> energy <strong>of</strong> a graph, Czechoslovak<br />
Math.Jour. 56(131) (2006), 1207-1213.<br />
Received: February, 2009